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Generalization of Queueing Network Product Form Solutions to Stochastic Petri Nets
February 1991 (vol. 17 no. 2)
pp. 99-107

A new solution is given for the steady-state probability computation of closed synchronized queuing networks. A closed synchronized queuing network is a particular Markov stochastic Petri net (a bounded and monovaluated Petri net with a strongly connected reachability graph and constant firing rates independent of markings). It is shown that the steady-state probability distribution can be expressed using matrix products. The results generalize the Gordon-Newell theorem. The solution is similar to the Gordon-Newell product form using matrix and vectors instead of scalars. A prototype solver developed from this result is presented.

[1] F. Baskett, K. M. Chandy, R. R. Muntz, and F. G. Palacios, "Open, closed, and mixed networks of queues with different classes of customers,"J. ACM, vol. 22, no. 2, pp. 248-260, 1975.
[2] G. W. Brams,Réseaux de Petri: Théorie et Pratique. Paris: Masson, 1983.
[3] J. P. Buzen, "Computational algorithms for closed queueing networks with exponential servers,"Commun. ACM, vol. 16, no. 9, Sept. 1973.
[4] H. Flatto and S. Hahn, "Two parallel queues created by arrivals with two demands,"SIAM J. Appl. Math., vol. 44, pp. 1041-1053, 1984.
[5] G. Florin and S. Natkin, "One place unbounded stochastic Petri nets: Ergodicity criteria and steady state solution,"J. Syst. Software, vol. 1, no. 2, pp. 103-115, 1986.
[6] G. Florin and S. Natkin, "Les réseaux de Petri stochastiques: Théorie, techniques de calcul, applications," Thèses de doctorat d'état, Univ. Paris VI, June 1985.
[7] G. Florin, C. Fraize, and S. Natkin, "Stochastic Petri Nets: Properties, applications, and tools," CEDRIC, Res. Rep. 90005; to be published inMicroelectron. Rel. (Special Issue on Petri Nets).
[8] W. J. Gordon and G. F. Newell, "Closed queueing systems with exponential servers,"Oper. Res., vol. 15, no. 2, pp. 252-267, 1967.
[9] P. Heidelberg and S. S. Lavenberg, "Computer performance evaluation methodology,"IEEE Trans. Comput., vol. C-33, no. 12, Dec. 1984.
[10] J. R. Jackson, "Job-shop like queueing systems,"Management Sci., vol. 10, pp. 131-142, 1963.
[11] M. Ajmone Marsan, G. Balbo, and G. Conte, "A class of generalized stochastic Petri nets for the performance evaluation of multiprocessor systems,"ACM Trans. Comput. Syst., vol. 2, pp. 93-122, May 1984.
[12] M. K. Molloy, "Performance analysis using stochastic Petri nets,"IEEE Trans. Comput., vol. C-31, no. 9, Sept. 1982.
[13] M. F. Neuts,Matrix Geometric Solutions in Stochastic Models. Baltimore, MD: Johns Hopkins University Press, 1981.
[14] J. L. Peterson,Petri Net Theory and the Modeling of Systems. Englewood Cliffs, NJ: Prentice-Hall, 1981.
[15] A. A. Lazar and T. G. Robertazzi, "Markovian Petri net protocols with product form solution," inProc. IEEE Infocom '87, Conf. Computer Communications. Washington, DC: IEEE Computer Society Press, 1987, pp. 1054-1062.
[16] M. Reiser and S. Lavenberg, "Mean value analysis of closed multichain queueing networks,"J. ACM, vol. 27, no. 2, Apr. 1980.
[17] J. M. Toudic and H. Alaiwan, "Recherche des semi-flots des verrous et des trappes dans les réseaux de Petri,"Technique et Science Informatique (TSI), vol. 4, no. 1, 1984.

Index Terms:
performance evaluation; queueing network product form solutions; stochastic Petri nets; steady-state probability; closed synchronized queuing networks; Markov stochastic Petri net; strongly connected reachability graph; constant firing rates; matrix products; Gordon-Newell theorem; performance evaluation; Petri nets; queueing theory
Citation:
G. Florin, S. Natkin, "Generalization of Queueing Network Product Form Solutions to Stochastic Petri Nets," IEEE Transactions on Software Engineering, vol. 17, no. 2, pp. 99-107, Feb. 1991, doi:10.1109/32.67591
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